Suppose a quantum version of gravity exists (which I don´t think). What will be the quantities in the uncertainty relation connected with this quantum gravity? Some time ago I sent a letter to Lee Smolin after which of course no answer came. But later I saw in an article that in his version (LQG) volume and area of space were the two quantities. Is this the case in all suggested types of quantized gravity?
Models based on string theory have consistent quantization of gravity. Within these , there is theoretical work carried out for generalizing the uncertainty principle for quantum gravity.
It generalizes the usual space momentum uncertainty,(formula 15)
This one examines an uncertainty in space time, formula 2.2.
It is part of ongoing theoretical research.